38 THE THEORY OF SCREWS. [41- 



41. The Canonical Co-Reciprocals. 



If all the six screws of a co-reciprocal system are to pass through the same 

 point, they must in general constitute a pair of screws of pitches 4- a and 



a on an axis OX, a pair of screws of pitches -1- b and 6 on an axis Y 

 which intersects OX at right angles, and a pair of screws of pitches + c and 



c on an axis OZ perpendicular to both OX and OF. 



It is convenient to speak of a co-reciprocal system thus arranged as a set 

 of canonical co-reciprocals. The three rectangular axes OX, OY, OZ we may 

 refer to as the associated Cartesian axes. 



If a, , 2 2 , ... a 6 be the six co-ordinates of a screw referred to the canonical 

 co-reciprocals, then the pitch is given in general by the equation 



p a = a Or - 2 2 ) + b (ot s 2 - a 4 2 ) + c ( 5 2 - 6 2 ). 



It must be remembered that in this formula we assume that the co-ordi 

 nates satisfy the condition | 35 



1 = (! + a a ) 2 + (a, + 4 ) 2 + ( 5 + a*) 2 - 



Of course this condition is not necessarily complied with when a.^ ,, ... or 

 some of them are infinite, as they are in the case of a screw of infinite 

 pitch 44. 



In general the direction cosines of the screw a are 



42. An Expression for the Virtual Coefficient. 



Let X , fjf, v be the direction cosines of the screw B (of pitch p e ) which 

 passes through the point x, y , z . Let X&quot;, p&quot;, v&quot; be the direction cosines of 

 the screw a (of pitch p a ) which passes through the point x&quot; , y&quot;, z&quot;. Then it 

 can easily be shown that the virtual coefficient of 6 and a is half the 

 expression 



j x - x&quot;, y - y\ z - z&quot; 



(l } 9 + P&amp;lt;t&amp;gt;) (X X&quot; + p fJi&quot; + v v&quot;) X , fJL , v 



X&quot; , /*&quot; , &quot;&quot; 



43. Equations of a Screw. 



Given the six co-ordinates a l , 2 , . . . 6 of a screw, with reference to a set 

 of six canonical co-reciprocals, it is required to find the equations of that screw 

 with reference to the associated Cartesian axes. 



If we take for 6 in the expression just written the screw of pitch a in the 

 canonical system, thus making 



X = 1 ; pf = ; v = ; x = ; y = ; z = 0, 



